M 454/455. Intro Dynamical Systems I-II

Mark Pernarowski
Nonlinear Dynamics and Chaos, Steven Strogatz, 2nd ed.
We will cover topics in the text in much greater detail.
The course % will determined as follows:
                         Percent       Date 
see below
Dates and details for Midterm and Final exam will be announced later.
Below the Homework and due dates will be posted.
            Due Date   
     Wed, Sept 19  (454)
   Wed, Oct 3
    HW3    Wed, Oct 17  (454)
     HW4    TBA  (454)
     HW5    TBA  (454)
     HW6    TBA  (454)
     HW7    TBA  (455)
     HW8    TBA
     HW9    TBA  (455)
     HW10    TBA  (455)
     HW11    TBA  (455)

 Periodically I will post handwritten and/or typeset lecture notes here:

     Notes_0        Dynamical system examples, overview, systems, ode, Maps, PDE's and terminology
   First order ODE: stability and phase portraits
     Notes_2a    First Order ODE: existence theorems, blowup, Picard iteration, Taylor Series
     Notes_2b    Linear Stability for fixed points
     Notes_3    Comparison Theorem and Potential Functions
     Notes_4    Saddle and Transcritical Bifurcations
     Notes_5    Pitchfork bifurcations and Structural Stability
     Notes_6    Taylor Series, Implicit Function Theorem, Hyperbolicity, Normal Forms
     Notes_7    Perturbed TC and PF bifurcations and Budworm Population Model
     Notes_8    Dynamics on S1 (unit circle) brief overview
     Notes_9    Planar system notation, fixed point and stability definitions
     Notes_10    Brief review of linear algebra
     Notes_11    Linear planar systems: saddles,nodes,spirals,centers,line of equilibria
     Notes_12    Linear planar systems: classification and manifolds E^s, E^c and E^u (and here)
     Notes_13    Planar systems: nullclines, linearization,hyperbolicity,flow, H-G Theorem
     Notes_14    W^s,W^u, homo/heteroclinic orbits, Lotka Volterra, Conservative & Hamiltonians
     Notes_15    Nonlinear centers and Hamiltonian systems, Reversible systems
     Notes_16     Index Theory
     Notes_17    Polar coord, limit cycles, periods, gradient sys., dissipation, Liapunov fns.
     Notes_18    Dulac Criterion
     Notes_19    Poincare-Bendixson: positive invariance, trapping regions, omega-limit sets
     Notes_19a    Poincare application to Glycolosis Oscillator model
     Notes_20    Relaxation Oscillations: Period estimates
     Notes_21    Bifurcations of Fixed points
     Notes_22    Hopf Bifurcations. Summary notes here
     Notes_23    Global Bifurcations of Periodic Orbits and Return Maps
     Notes_24    Map examples, cobwebs, fixed points, stability, linear stability, hyperbolicity
     Notes_24.5    Saddle Node bifurcations of fixed points
     Notes_25    Logistic Map: fixed point and period 2 orbit existence/stability
     Notes_26    Logistic Map: mu=4 (period 2,3,...) orbit existence
     Notes_27    Period Doubling Cascade and numerical x vs. mu bifurcations
     Notes_28    Logistic Map: Exact solutions and periodicity = rational numbers
     Notes_29    Intro to Itineraries and Transition graphs (updated)
     Notes_30    Allowable paths and shared itineraries
     Notes_31    Sensitivity on Init Cond, Lyapunov exponents, chaotic orbits
     Notes 32    Conjugate maps
     Notes_33    Period 3 orbits, Cantor sets, Bit-Shift map
     Notes_34    Introduction to Planar Maps: Defns, Translation and Scaling Maps
     Notes_35    Area preserving maps, linear explicit solutions, Stability intro.
     Notes_36    Area preserving: detDf=1, linear cases, Henon map.
 Complex eigenvalues, pure rotation, hyperbolicity, linearization 
 Henon map with homoclinic entanglement, strange attractors

Here are some supplementary notes I may refer to Book.pdf


          Midterm 1 -  Friday October 26 (in class)

You may bring a single (2-sided) sheet of notes - handwritten or typed. 

Page 6 of Note 6 on Normal forms will be appended to the exam but you will still need to know the two variable Taylor Series expansions for Normal Forms.


HW1-HW3, Posted Notes1-6 covers more than you need. Noteable exclusions:

  • Picard Iteration in HW 1
  • Comparison Theorems in Note 3
  • Structural stability and 2-parameter bifurcation diagrams in Note 7
3)   You won't need scrap paper. You will answer questions on the exam iteself.

You'll need to know basic definitions for the various forms of stability, for

isolated fixed points, hyperbolic, TC, SN, PF, etc.

    Final Exam (Takehome) Due: TBA     here

Topics that may be on the exam:

    Posted Notes 16-23 (inclusive) covering HW7-9

Of particular note:

  • Index Theory but not the line integral formula
  • Conversion to polar coordinates
  • Gradient systems
  • Dulac Criteria
  • Liapunov functions
  • positive invariance, trapping regions, omega-limit sets
  • Poincare Bendixson Theorem(both)
  • Relaxation Oscillations (fast-slow period estimate)
  • Hopf Bifurcations.
  • Global Bifurcations of Periodic Orbits/Fixed points in Polar Coordinates
1)          You may only use the text, your class notes or the online notes.
2)   You may use software for drawing functions curves or level sets
3)   You may not talk to any fellow students
4)   You may ask me to clarify questions
    • Dynamics on R
      • fixed points, stability, linear stability
      • Existence, Uniqueness, Picard Iteration, Taylor series methods
      • Blowup, Comparison arguments
    • Elementary Bifurcations on R
      • Saddle Node, Transcritical, Pitchfork bifurcations
      • Normal Forms
      • Stability Diagrams
    • Dynamics on S^1
      • Phase portraits, periodic orbits
      • Bifurcations
      • Asymptotic Period estimates and bottlenecks
      • Compactification of dynamics on R onto S^1
    • Linear Planar Systems
      • Phase portraits, fixed points
      • Fundamental matrix solutions
      • Stability in from Tr(Df) and det(A)
      • linear manifolds E^s, E^c and E^u
    • Planar Systems Introduction
      • Nullclines, Flow, Hyperbolic fixed points
      • Homeomorphisms and Topological Equivalence
      • Liapunov Stable, Attracting, isolated, asymptotically stable
      • Hartman-Grobman Theorem
    • Special Structures in Planar Systems
      • Conservative Systems
      • Hamiltonian Systems
      • Reversible Systems
      • Theorems for Nonlinear Centers
    • Index Theory
      • Definition and Calculation
      • Integral formulation
      • Key properties and Theorems
      • Proving nonexistence of Periodic orbits
    • Periodic Orbits in Planar Systems
      • Conversion to Polar coordinates
      • Limit Cycles and Stability Definitions
      • Gradient Systems - no periodic orbits
      • Dissipative systems - no periodic orbits
      • Liapunov Functions - no periodic orbits
      • Dulac Criterion
      • Poincare-Bendixson Thms (trapping regions)
      • Omega Limit sets, general Poincare-Bendixson.
      • Poincare Return Maps
    • 1-D Maps
      • Euler's method, Newton's method, tent and logistic maps
      • fixed points, stability, asymptotic stability, orbits, cobwebs
      • Linear stability: taylor series and rigorous proofs
      • Introductory examples including superstability
      • Logistic Map: fixed points, period 2, period 3,...
      • x



Upon completion of this course, a student will be able to: Provide a qualitative bifurcation analysis of a simple one-dimensional, one parameter nonlinear differential equation; Understand and analyze basic types of linear and nonlinear oscillators; Linearize a two-dimensional non-linear system of differential equations at an equilibrium, and use this linearization to analyze the behavior of nearby solutions; Analyze dynamics of a two-dimensional nonlinear system of differential equations using a phase plane analysis.

Upon completion of this course, a student will be able to: Find fixed points and low period periodic points for simple one-dimensional maps both graphically and analyticaly; Analyze dynamics of one dimensional maps using symbolic dynamics; Understand and be able to reproduce construction of the Smale's horseshoe; Have an understanding of simple models of chaotic dynamics.